Theorem subrng0

Description: A subring always has the same additive identity. (Contributed by AV, 14-Feb-2025)

	Ref	Expression
Hypotheses	subrng0.1	$$⊢S=R{\upharpoonright }_{𝑠}A$$
	subrng0.2	$$⊢\underset{\dot{}}{0}=0_R$$
Assertion	subrng0	Could not format assertion : No typesetting found for |- ( A e. ( SubRng ` R ) -> .0. = ( 0g ` S ) ) with typecode |-

Proof

Step	Hyp	Ref	Expression
1		subrng0.1	$$⊢S=R{\upharpoonright }_{𝑠}A$$
2		subrng0.2	$$⊢\underset{\dot{}}{0}=0_R$$
3		subrngsubg	Could not format ( A e. ( SubRng ` R ) -> A e. ( SubGrp ` R ) ) : No typesetting found for |- ( A e. ( SubRng ` R ) -> A e. ( SubGrp ` R ) ) with typecode |-
4	1 2	subg0	$$⊢A\in \mathrm{SubGrp}\left(R\right)\to \underset{\dot{}}{0}=0_S$$
5	3 4	syl	Could not format ( A e. ( SubRng ` R ) -> .0. = ( 0g ` S ) ) : No typesetting found for |- ( A e. ( SubRng ` R ) -> .0. = ( 0g ` S ) ) with typecode |-